# mclaurin series and taylor inequality

• May 7th 2010, 08:55 AM
twofortwo
mclaurin series and taylor inequality
use mclaurin series to compute the sin 3 degrees, correct to 5 decimal places.
use taylos inequality to show you are within 5 decimal places.

i found the bound for the sinx, its obviously one but i dont know wat that means or how to set it up.

thanx!
• May 7th 2010, 09:01 AM
twofortwo
Rn, remainder and estimating sum
thanx for all help!
• May 9th 2010, 05:53 PM
twofortwo
• May 9th 2010, 06:04 PM
AllanCuz
Quote:

Originally Posted by twofortwo
use mclaurin series to compute the sin 3 degrees, correct to 5 decimal places.
use taylos inequality to show you are within 5 decimal places.

i found the bound for the sinx, its obviously one but i dont know wat that means or how to set it up.

thanx!

The McLaurin series is

$\displaystyle \sum_{k=0}^{ \infty } = \frac{ f^{ (k) } (0) } {k!} x^k$

It is well known that the Mclaurin series of sinx leads to...

$\displaystyle sinx = \sum_{n=0}^{ \infty } \frac{ (-1)^n }{(2n+1)!} x^{2n + 1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}....$

I'm assuming your questions says evaluate until we get to the third power (not entirely clear from your post) so what you're looking for is

$\displaystyle x - \frac{x^3}{3!}$

But to be honest, your question is too unclear for me to get to any sort of answer here. What are you looking for?
• May 9th 2010, 06:16 PM
twofortwo
sorry, thats exactly why im so confused cause thats how the question is worded

i think they are asking to find the sin of 3 degrees using the inequality correct to decimal places

thanx
• May 9th 2010, 06:19 PM
TheEmptySet
Quote:

Originally Posted by twofortwo
sorry, thats exactly why im so confused cause thats how the question is worded

i think they are asking to find the sin of 3 degrees using the inequality correct to decimal places

thanx

You do not need to use Taylors ineqality beucase it is an alternating series. The error in an alternating series is always less than the value of the next term in the series.
• May 10th 2010, 03:05 AM
HallsofIvy
Taylor's inequality says that the error in approximating the infinite series (Taylor's series) $\displaystyle \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x- x_0)^n$ by truncating at the Nth power is less than $\displaystyle \frac{M}{(N+1)!}|x- x_0|^{N+1}$ where M is an upper bound on the absolute value of the Mth dervivative of f between x and $\displaystyle x_0$.

All derivatives of sin(x) are $\displaystyle \pm sin(x)$ or $\displaystyle \pm cos(x)$ both of which have maximum value over all x of 1, as you say. Of course, the derivative of sin(x) is cos(x) and the derivative or cos(x) is -sin(x) only when x is measured in radians so $\displaystyle x_0= 0$ and $\displaystyle x= \frac{3}{180}\pi= 0.03491$, approximately.

So the errror is less than $\displaystyle \frac{1}{(N+1)!}(.03491)^{N+1}$ and you want to make that .00001.

I would recommend just evaluating that at N= 3, 4, 5, ... until it gets small enough.